Extensions of homogeneous coordinate rings to $A_{\infty}$-algebras

We study $A_{\infty}$-structures extending the natural algebra structure on the cohomology of $\oplus_{n\in\mathbb{Z}} L^n$, where $L$ is a very ample line bundle on a projective $d$-dimensional variety $X$ such that $H^i(X,L^n)=0$ for $0 > i > d$ and all $ n \in \mathbb{Z}$. We prove that there exists a unique such nontrivial$A_{\infty}$-structure up to a strict $A_{\infty}$-isomorphism (i.e., an $A_{\infty}$-isomorphism with the identity as the first structure map) and rescaling.

In the case when $X$ is a curve we also compute the group of strict $A_{\infty}$-automorphisms of this $A_{\infty}$-structure.

Keywords

$A_{\infty}$-algebra, $A_{\infty}$-isomorphism, homogeneous coordinate ring

2010 Mathematics Subject Classification

18E30, 55P43

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Published 1 January 2003